Recurring Decimals

Why this matters

Some fractions have neat decimals ( $\frac{3}{8} = 0.375$ ). Others, like $\frac{1}{3}$ , go on forever. Knowing which is which — before you start dividing — tells you whether a calculator answer is exact or has been rounded, and whether a decimal answer can ever be exact at all.

Why only 2s and 5s terminate

A terminating decimal is really a fraction over 10, 100, 1000, … and $10 = 2 \times 5$ . You can only turn a denominator into a power of 10 by multiplying it up, which is possible only if it is already made of 2s and 5s:

$\tfrac{7}{40} = \tfrac{7}{2^3 \times 5} = \tfrac{7 \times 25}{2^3 \times 5 \times 25} = \tfrac{175}{1000} = 0.175$

A denominator with a 3, 7, 11 or any other prime can never become a power of 10.

Calculator displays

A calculator might show $\frac{106}{333}$ as $0.31831831832$ . The last digit is rounded, which hides the pattern. The true value is $0.\dot{3}1\dot{8}$ .

Fraction	Decimal	Terminates?
$\frac{3}{12} = \frac{1}{4}$	$0.25$	Yes ( $4 = 2^2$ )
$\frac{7}{40}$	$0.175$	Yes ( $40 = 2^3 \times 5$ )
$\frac{2}{25}$	$0.08$	Yes ( $25 = 5^2$ )
$\frac{5}{24}$	$0.208\dot{3}$	No ( $24 = 2^3 \times 3$ )
$\frac{5}{7}$	$0.\dot{7}1428\dot{5}$	No

Simplify first! $\frac{3}{12}$ has a 3 in its denominator, but it simplifies to $\frac{1}{4}$ , which terminates. Always cancel before you look at the primes.

Key rules and formulas

Dot notation

$\frac{2}{9} = 0. \dot{2} \frac{517}{999} = 0. \dot{5} 1 \dot{7} \frac{2}{15} = 0.1 \dot{3}$

A dot over a digit means it repeats forever. If a block of digits repeats, put dots over the first and last digits of the block.

Terminating or recurring?

Write the fraction in its simplest form, then write the denominator as a product of primes. If the only primes are 2s and 5s, the decimal terminates. Any other prime and it recurs.

Ninths and ninety-ninths

$\frac{1}{9} = 0. \dot{1} \frac{1}{99} = 0. \dot{0} \dot{1} \frac{1}{999} = 0. \dot{0} 0 \dot{1}$

\frac{7}{9} = 0. \dot{7}

\frac{23}{99} = 0. \dot{2} \dot{3}

and

\frac{517}{999} = 0. \dot{5} 1 \dot{7}

— the repeating block goes on top.

Back to a fraction

Reverse it: a single repeating digit goes over 9, a repeating pair over 99, a repeating three over 999. Then simplify:

0. \dot{3} \dot{6} = \frac{36}{99} = \frac{4}{11}

Worked examples

Worked example 1

Fraction to recurring decimal

Write

\frac{2}{15}

as a decimal.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Will it terminate?

Decide whether

\frac{14}{80}

and

\frac{10}{48}

give terminating decimals.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Using a known fact

Given

\frac{44}{90} = 0.4 \dot{8}

, write

\frac{22}{90}

and

\frac{11}{90}

as recurring decimals.

Try it yourself first, then reveal the solution one step at a time.

Fraction	Simplest form	Denominator as primes	Terminating or recurring?
14/80	Row 1, Simplest form	2³ × 5	Row 1, Terminating or recurring?
10/48	Row 2, Simplest form	2³ × 3	Row 2, Terminating or recurring?
8/100	Row 3, Simplest form	5 × 5	Row 3, Terminating or recurring?
6/33	Row 4, Simplest form	11	Row 4, Terminating or recurring?

Fraction

Simplest form

Denominator as primes

Terminating or recurring?

14/80

2³ × 5

10/48

2³ × 3

8/100

5 × 5

6/33

In this lesson you will learn to…

Why this matters

Why only 2s and 5s terminate

Calculator displays

Key rules and formulas

Dot notation

Terminating or recurring?

Ninths and ninety-ninths

Back to a fraction

Worked examples

Fraction to recurring decimal

Will it terminate?

Using a known fact

Practice: recurring decimals